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Sosa, KishoreYalamanchilli State Street Research & Management CIFEr, New York March 27th, 2000tS!>#B+gRisk-constrained Optimization Facilitates Integration of Various Sector Views In Portfolio Constructionhg AgendakProblem Statement Why Multi-level Optimization? Proposed Multilevel Risk-Constrained Optimization Algorithm Sub-level: Nondollar Sectors vs. Domestic Index Top Level: Domestic Sectors + Customized Nondollar Portfolio Brief Review of Our Past Research Sector Structure Risk Model for G-13 Nondollar Government Bond Markets Sample Optimization Results Conclusion DlmlmD-0Risk-Constrained Optimization Problem Definition10H/6Problem Definition (Cont d)I0Why Multi-Level Optimization P3*Proposed Multilevel Optimization Algorithm+*$Q4fProposed Multilevel Optimization Algorithm (Cont d)44,!Brief Review of Our Past Research""$;Risk Models Used Rolling GARCH-PJ Sample Domestic Results R5(Methodology Requirements for Risk Models)( RAccuracy Nonstationary Non-normal Robustness Feasible automation and maintenance 6 0 0 < Risk Models Used5Rolling Variance-Covariance G-13 Government Bond Yields Investment Grade Corporate and ABS OASs GARCH and Garch-t Mortgage passthroughs G-13 currencies Garch-PJ Used for high yield and emerging markets Univariate GARCH with Persistent Jumps Rolling white noise correlation matrix Exogenized jump frequenciesD' *jC%*; #, C  Rolling GARCH-PJ (univariate)  A variation of GARCH(1,1) that features Bernoulli-style jumps Dst = a0 + et, where et = sqrt(ht)ut + jt, with ut ~ N(0,1) i.i.d. ht = g0 + g1 e2t-1 + g2ht-1 jt ~ N(mj,sj2) with probability p 0 with probability 1-p Jump occurrences in this model will induce a volatility spike in subsequent days Bernoulli, rather than Poisson jumps, simplify and speed up the parameter estimation procedure VaR estimates are also produced via simulation Jump frequencies are also allowed to depend on exogenous or past data>%>                      S?-X         Model ChoiceThe skewness and kurtosis of the standardized innovations support GARCH-PJ Brazil 1992-1999: Skewness Kurtosis Rolling Var-Covar 5.94 99.67 GARCH 2.96 47.20 GARCH-PJ * 0.16 3.50 GARCH-PJ Exo* 0.12 3.42 *jump days excluded0Lb Q b."Multivariate ARCH IssuesM1@Sample Domestic Results (Cont d)!! 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(V(   (r ( S C    ( TDgֳgֳ ?0  m :  ( TdDgֳgֳ ?` '-Sub-level: Optimize Relative Return of Nondollar Sectors to Domestic Index (Decision Variables: NWi, Cwi ) Subject to: Relative CVaR Limit Allotted to Nondollar Example (3-Month, 97.5% Confidence Level): Total Relative CVaR =100bp Allotted Nondollar CVaR at Total Portfolio Level = 30bp Allotted Nondollar Relative CVaR at Sector Level = 300bp (All numbers measured w.r.t. domestic index) Hedge limits (i.e., 0.95 <[CWi/ NWi ]< 1.05) Scenario constraints Other constraints (related to guideline or research view) Country weight or duration contribution  l*+.}( U    *+-      P(Pa:H ( 0޽h ? ___f3f  E=,(  ,r , S $E`    , TEgֳgֳ ?0  m : * , TEgֳgֳ ?  rTop-level: Optimize Relative Return vs. Overall Index with Decision Variables as Domestic Sector Weights (DWi) Weight of Customized Nondollar Portfolio (Nwcustomized index ) Specific opportunistic countries can be segregated and optimized at the top level (e.g. Greece) Subject to: Relative CVaR Limit vs.. Overall benchmark Scenario constraints Traditional constraints F]lX F  ,  l-,%DH , 0޽h ? ___f3f  *"$( |Lv4 $ $  fDFgֳgֳ ?P`p    $ # lFgֳgֳ ?0   H $ 0޽h ? ___f3f  6.00( b` 0 0 # ldGgֳgֳ ?P`p    0 3 rGgֳgֳ ?0   H 0 0޽h ? ___f3f  *"P,( ֳ , ,  fHgֳgֳ ?    , # lDIgֳgֳ ?0   H , 0޽h ? ___f3f   0(p\(  \ \ # lKgֳgֳ ?    \ # ltgֳgֳ ? `  H \ 0޽h ? ___f3fW  .W&WyytV(  t t # l4gֳgֳ ?   X t 0?YTjB t B1??ojB t B1?Y  pjB t B1?YuupjB t B1?YpjB t B1?YDDpjB  t B1??ojB  t B1?YpjB  t B1?Y{{pjB  t B1?YpjB  t B1?YJJpjB t B1??ojB t B1?YpjB t B1?YpjB t B1?YpjB t B1?YQQpjB t B1??ojB t B1?Y  pjB t B1?YpjB t B1?YpjB t B1?YWWpjB t B1??ojB t B1?Y' ' pjB t B1?Y pjB t B1?Y pjB t B1?Y^ ^ pjB t B1?? ojB t B1?Y- - pjB t B1?Y pjB  t B1?Y pjB !t B1?Yd d pjB "t B1?? ojB #t B1?Y4 4 pjB $t B1?Y pjB %t B1?YpjB &t B1?YkkpjB 't B1??ojB (t B1?Y::pjB )t B1?YpjB *t B1?Y  pjB +t B1?YqqpjB ,t B1??ojB -t B1?YAApjB .t B1?YpjB /t B1?YpjB 0t B1?YxxpjB 1t B1??ojB 2t B1?YGGpjB 3t B1?YpjB 4t B1?YpjB 5t B1?Y~~pjB 6t B1??ojB 7t B1?YMMpjB 8t B1?YpjB 9t B1?YpjB :t B1?YpjB ;t B1?pq t TT?xp 650 ?t T?!p 675 @t T?hD p 7100 At Tt?n J p 7125 Bt T?u Q p 7150 Ct T4?{Wp 7175 Dt T?^p 7200 Et T?dp 7225 Ft TT?kp 7250pB Gt H1?jB Ht B1?./jB It B1?jB Jt B1?jB Kt B1?./jB Lt B1? jB Mt B1?W X jB Nt B1? jB Ot B1? jB Pt B1?  jB Qt B1? jB Rt B1?= > jB St B1? jB Tt B1?f g jB Ut B1? jB Vt B1? jB Wt B1? jB Xt B1?$ % jB Yt B1?jB Zt B1?MNjB [t B1?jB \t B1?jB ]t B1?vwjB ^t B1?  jB _t B1?jB `t B1?34jB at B1?jB bt B1?\]jB ct B1?jB dt B1?jB et B1?jB ft B1?jB gt B1?jB ht B1?BCjB it B1?jB jt B1?lmjB kt B1?jB lt B1?jB mt B1?p nt T?/ 6-3 ot T?   6-2 pt Tt?   6-1 qt T?| 50 rt T4?c 51 st T?I 52 tt T?/ 53  ut  BdC>DE$F,1?  *>~Sh|#7Lau0EZn)>Sg|   $# /".7.K=`;u;3266421-/DYn !!(=Rg{ ! 6 K_ t  /CXm ' < Q f {         5 J _ s         . C W l         ' < P e z    Z ] U ,  5 I ^ s      -qBqWpkNL "'*,()&/;$Pdy % %,4HI]r("- B'V)k &; Ody&@:@<A =836H#] q ,$A*V+j,+*l%:Oc@  vt  BdCDE$F,1?  59*9>6SFhV|R#7Lau0EZn  )>Sg!|00:CNWX h"k7iK`}u|rtxywvtn/WDWYWnVVYZZWWWW(V=VRSgE{EEDJIF? ?!?6>K:_;t>>   /CXm ' < Q f {         5 J _ s         . C W l         ' < P |e z s P S      5 I ^ s       -BWkF>A??;<&@;2P"dy  & &,4MI]ryjb-BVkpm^2B>D&C;GOAd@y_r 3H]mqe_c`WZar{,AVj}aXC%:O}c@  wt  BdC DE$F,1?  b*>gSh|br#g7LauBRb7W0xEZ7n=xG'gBbbb)]>LS=g|'r]x} ""7K,`u'rx,',/DYGn}2mB72R2(b=Rgr{xW'G =!"6KB_]tx`B/CXmmW{] ' < Q f r{  m       5 J _ s G     x }  . ,C W l  m  L W    7' < P e ;z &        5 I ^ Gs A  v b q L a-BRW} k e?D&;cPdAy j4.I]r*U_b-B_VkW59 'z&;ORdy( R3H]q=r,AV$jT=3# %:ROcA@l xt TT ?|8g  uABrazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99BB yt T!?%2  #and 90%&99% Var-Covar VaR estimates$$, H t 0޽h? ? ___f3fZ   hZ`Z|Y(  | | # l!gֳgֳ ?   X | 0?YTjB | B1?7gjB | B1?OgjB | B1?OvvgjB | B1?OgjB | B1?OFFgjB  | B1?7gjB  | B1?OgjB  | B1?O}}gjB  | B1?OgjB  | B1?OKKgjB | B1?7gjB | B1?OgjB | B1?OgjB | B1?OgjB | B1?ORRgjB | B1?7gjB | B1?O!!gjB | B1?OgjB | B1?OgjB | B1?OXXgjB | B1?7gjB | B1?O' ' gjB | B1?O gjB | B1?O gjB | B1?O^ ^ gjB | B1?7 gjB | B1?O- - gjB | B1?O gjB  | B1?O gjB !| B1?Od d gjB "| B1?7 gjB #| B1?O3 3 gjB $| B1?O gjB %| B1?OgjB &| B1?OjjgjB '| B1?7gjB (| B1?O99gjB )| B1?OgjB *| B1?O  gjB +| B1?OppgjB ,| B1?7gjB -| B1?O@@gjB .| B1?OgjB /| B1?OgjB 0| B1?OwwgjB 1| B1?7gjB 2| B1?OEEgjB 3| B1?OgjB 4| B1?OgjB 5| B1?O||gjB 6| B1?7gjB 7| B1?OLLgjB 8| B1?OgjB 9| B1?OgjB :| B1?OgjB ;| B1?hi <| T4"?g 50 =| T"?sg 625 >| T"?yg 650 ?| TT#?!g 675 @| T#?hD g 7100 A| T$?o K g 7125 B| T?u Q g 7150 C| TD?{Wg 7175 D| T?]g 7200 E| T?cg 7225 F| Td?ig 7250pB G| H1?  jB H| B1?WXjB I| B1?jB J| B1? jB K| B1?WXjB L| B1? jB M| B1? jB N| B1?G H jB O| B1? jB P| B1? jB Q| B1?6 7 jB R| B1? jB S| B1? jB T| B1?& ' jB U| B1? jB V| B1? jB W| B1?p q jB X| B1?  jB Y| B1? jB Z| B1?_ ` jB [| B1?  jB \| B1?  jB ]| B1?jB ^| B1?OPjB _| B1?jB `| B1?jB a| B1?>?jB b| B1?jB c| B1?jB d| B1?./jB e| B1?jB f| B1?xyjB g| B1?xyjB h| B1?jB i| B1?jB j| B1?ghjB k| B1? jB l| B1?jB m| B1?WXjB n| B1?jB o| B1?jB p| B1?FGjB q| B1?jB r| B1?jB s| B1?h t| T?X 6-3 u| T$? ,  6-2 v| T? f  6-1 w| T?   50 x| TD?? 51 y| T?y 52 z| T?M 53 {| Td? 54  ||  B`CDE$F,1?  k}*j>aSahj|hpinuiw o"o7L`~u0DYn(=Rf{ !6J_t.CXl'<Pez  4I^s^llvso-pB~Vvkdvn ~& r: _O Yd `y h l z  u u k z t3 H |] yq   v m { |   , @ U j         % 9 @N c w       M q 2 G [ p / + [ l ~ *?(T#i}Sel#8Mav+ki1EZo)>Sg| "7K `"u2 / D Yn%(+=Q)f{ ~ !=5Jo_|@  }|  B`CDE$F,1?  AO*<>Sh2|@J?9;? I"M7mLm`Suqqzo0DYn(=Rf{ !6J_t.CXl'<Pez  4I^s-?Lb=Q-@B[VSk2MilepH ^& 7: 2O !d !y ; 8 G e W @ = ; \3 [H Y] :q O U _ E M X u e , @ U tj  t   k   ~ r% 9 N c w  F N      2 +G J[ rp   4   1 *s?T%i} sz#'8[MKa%vmZ1EZo=XCgh0)>Sg| .$F v"y7K`u]-]t/tDYn/651T:(=Qf{ !5J_@  ~|  B`C DE$F,1?  e*>iSh|e r"i7L`%uJWe@\0wD!Y@nEwN3iJeee(a=SREf{3raw| .!6J7_t3rw737.CXNl|<nJ*@<W<'e<%Perzw\3N E .4 IJ^as9wJ-BVk%n\%a & : O d ry  n       *3 H ] q N     w |  *, 7@ U j  n  S \    @% 9 N %c kw   q   H p  2 $G [ Np   ' e  S Q*?WT i}B$#8M1avQD1EqZo]Y;e)>;Slg\|g'3 Q"7KW`u>+W/9DYnEr (=Q f1{Eh   !5WJ_@d | T?w N uABrazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99BB | T?>}  X$and 90%&99% GARCH(1,1) VaR estimates%%H | 0޽h ? ___f3fE[  ZZmZ( ]Dt    # lDgֳgֳ ?   X  0?YTjB  B1??ojB  B1?Y  pjB  B1?YuupjB  B1?YpjB  B1?YDDpjB  B1??ojB  B1?YpjB  B1?Y{{pjB  B1?YpjB  B1?YJJpjB  B1??ojB  B1?YpjB  B1?YpjB  B1?YpjB  B1?YQQpjB  B1??ojB  B1?Y  pjB  B1?YpjB  B1?YpjB  B1?YWWpjB  B1??ojB  B1?Y' ' pjB  B1?Y pjB  B1?Y pjB  B1?Y^ ^ pjB  B1?? ojB  B1?Y- - pjB  B1?Y pjB  B1?Y pjB ! B1?Yd d pjB " B1?? ojB # B1?Y4 4 pjB $ B1?Y pjB % B1?YpjB & B1?YkkpjB ' B1??ojB ( B1?Y::pjB ) B1?YpjB * B1?Y  pjB + B1?YqqpjB , B1??ojB - B1?YAApjB . B1?YpjB / B1?YpjB 0 B1?YxxpjB 1 B1??ojB 2 B1?YGGpjB 3 B1?YpjB 4 B1?YpjB 5 B1?Y~~pjB 6 B1??ojB 7 B1?YMMpjB 8 B1?YpjB 9 B1?YpjB : B1?YpjB ; B1?pq < T?p 50 = T?rp 625 > Td?xp 650 ? 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C W l         ' < lP Ae Yz n  9   .   5 I ^ s D D     S-ABnWk-k0y&;PduyN.> $`4;IN]/rUlpl2-BAV=k?FHW`]~&f;OdyXc,2 cs3Hd]@q6Ag>`x,AVj  %:FOcx@   ~  BdCQDE$F,1?  *Y>lS[hP|meqv{H#7nLau50$EZnX(4!vAE+)>=ShgP|%(#UH "7K#`u<!/DYny#$'(=R'g{   !(6K_t &*QX7/CdX?mtYn/ 8' n< eQ Lf c{ v   ]  o  n 5 J |_ hs    L i    . C W l  O  .     ' N< zP le z  F   1 +   u5 |I n^ s      f 8-B|W kak[v^&k;Pdy # ;4Im]Br-0BoV7k~gv&;OdFyt1a z3H]q#AiW,(AVj(%F:O~c@    BdCg DE$F,1?  D*>ISh|cDR#I7_Laullh*7Du!;0VEZ!n%V.I*DDD)@>2S%g|hR@V[_ "~7_K`uuRV/pDlY.n[M* !7(D=RgR{Vl;. %!6K*_@tc _lVo*/CXmM;@ ' < Q f R{  M    l ~ y 5 J _ _s .   y  V [ c . C W l  M ~ 2 ;    !' < P e Pz   \   5 g  5 I ^ .s    D  2 P-B7Wz kc&;Pdy=h l4IJ]rCf,D-BVEk;L+&;O7dy$n 73 H]qic%R,AVj %Swjf %j:7Oc@m  TQ?>m ) uABrazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99BB  TR?280  ['and 90%&99% GARCH-PJ(1,1) VaR estimates((H  0޽h ? ___f3f{[  [ [Z(    # lTSgֳgֳ ?   X  0?YTjB  B1??ojB  B1?Y  pjB  B1?YuupjB  B1?YpjB  B1?YDDpjB  B1??ojB  B1?YpjB  B1?Y{{pjB  B1?YpjB  B1?YJJpjB  B1??ojB  B1?YpjB  B1?YpjB  B1?YpjB  B1?YQQpjB  B1??ojB  B1?Y  pjB  B1?YpjB  B1?YpjB  B1?YWWpjB  B1??ojB  B1?Y' ' pjB  B1?Y pjB  B1?Y pjB  B1?Y^ ^ pjB  B1?? ojB  B1?Y- - pjB  B1?Y pjB  B1?Y pjB ! B1?Yd d pjB " B1?? ojB # B1?Y4 4 pjB $ B1?Y pjB % B1?YpjB & B1?YkkpjB ' B1??ojB ( B